We now prove the theorem. By quantum stein's lemma [1]} and quantum Sanov's theorem [2]} we have
\(\begin{aligned}\underset{n\rightarrow \infty }{\lim }\; \frac{1}{n}\log (1- P^n_D(\Pi ^n_1))=-\underset{\begin{array}{c}\rho ^{(n)}_1\in \mathcal {R}\end{array}}{\inf }S(\rho ^{(n)}_1\Vert \rho ^{\prime }_1),\end{aligned}\) 
